Spatial synchrony in population dynamics: The effects of demographic stochasticity and density regulation with a spatial scale

Accepted Manuscript

Spatial synchrony in population dynamics: the effects of demographic stochasticity and density regulation with a spatial scale Steinar Engen, Bernt-Erik Sæther PII: DOI: Reference:

S0025-5564(16)00023-7 10.1016/j.mbs.2016.01.012 MBS 7747

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Mathematical Biosciences

Received date: Revised date: Accepted date:

3 June 2015 2 December 2015 7 January 2016

Please cite this article as: Steinar Engen, Bernt-Erik Sæther, Spatial synchrony in population dynamics: the effects of demographic stochasticity and density regulation with a spatial scale, Mathematical Biosciences (2016), doi: 10.1016/j.mbs.2016.01.012

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Highlights • The spatial scale of population synchrony is analyzed by generalizing previous results.

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• Demographic noise interferes with environmental noise in generating

fluctuations in population density in space. A characteristic area A0 within which demographic noise interferes with environmental noise, as

well as a dimension free spatial demographic coefficient expressing the

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relative effect of the two types of noise, are defined.

• The spatial scale at which density regulation operates affects the spatial synchrony. An increase of the scale may produce larger synchrony at

small distances and smaller and even negative values at larger distances.

statistical analysis.

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• The observed spatial scale depends on the sampling areas used in the

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• General scaling results for the spatial autocorrelation are derived when density regulation with a spatial scale, dispersal, spatially autocorre-

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lated environmental noise as well as demographic noise are included.

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Spatial synchrony in population dynamics: the effects of demographic stochasticity and density regulation with a spatial scale

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Steinar Engen1 and Bernt-Erik Sæther2 Centre for Biodiversity Dynamics, Department of Mathematical Sciences,

Norwegian University of Science and Technology, NO-7491 Trondheim, Nor-

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way

Centre for Biodiversity Dynamics, Department of Biology, Norwegian Uni-

versity of Science and Technology, NO-7491, Trondheim, Norway

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Engen:+47 90635053, [email protected]

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Sæther: [email protected]

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Running head: spatial synchrony

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ABSTRACT

We generalize a previous simple result by Lande et al. [29] on how spatial au-

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tocorrelated noise, dispersal rate and distance as well as strength of density regulation determine the spatial scale of synchrony in population density. It is shown how demographic noise can be incorporated, what effect it has on variance and spatial scale of synchrony, and how it interacts with the point process for locations of individuals under random sampling. Although the

effect of demographic noise is a rather complex interaction with environmen-

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tal noise, migration and density regulation, its effect on population fluctuations and scale of synchrony can be presented in a transparent way. This is

achieved by defining a characteristic area dependent on demographic and environmental variances as well as population density, and subsequently using

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this area to define a spatial demographic coefficient. The demographic noise acts through this coefficient on the spatial synchrony, which may increase or decrease with increasing demographic noise depending on other parameters.

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A second generalization yields the modeling of density regulation taking into account that regulation at a given location does not only depend on the den-

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sity at that site but also on densities in the whole territory or home range of individuals. It is shown that such density regulation with a spatial scale

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reduces the scale of synchrony in population fluctuations relative to the simpler model with density regulation at each location determined only by the

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local point density, and may even generate negative spatial autocorrelations.

Key words: spatial population dynamics, density regulation, environmental noise, demographic noise, dispersal, spatial scale.

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1. Introduction In population dynamics spatial synchrony is measured by parameters expressing how correlations in population size or their temporal fluctuations

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changes with distance between sites. Such correlations are of great interest because they in general affect the dynamics and in particular extinction risks [2, 6, 11, 23, 30, 42]. Populations can be sychronized by the effects of spatially

correlated environmental variables [5, 28, 33, 42] such as weather [18], by bi-

ological interactions like predation [24, 47] as well as dispersal rates and distances [16, 43]. These effects may differ considerably among species [30, 39],

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leading to large interspecific differences in the spatial scale of population synchrony [43].

Spatial population dynamics was first stydied by metapopulation models introduced by Levins [31, 32], and then extended especially by Hanski and col-

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laborators [21, 22] . Later models in continuous time and space have proved useful in analyzing how environmental noise, density regulation and disper-

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sal rates in general influence population synchrony [10, 12, 27, 29, 30] that in turn may have considerable effects on extinction processes [11].

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In the absence of migration, linear spatial population models (or log population size models) in discrete or continuous time have the simple property,

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known as the Moran effect [36], that the spatial autocorrelation of population size (or log population size) equals the spatial autocorrelation in the noise,

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provided that the strength of density regulation is constant in space [12]. However, spatial dynamic models with migration has shown that dispersal may have a substantial additional effect on the scale [10, 29]. In particular, under weak local density regulation even small migration over short distances may lead to a spatial scale of population synchrony that is much larger than the scale of the environmental noise. Lande et al. [29] expressed this by a

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simple result, using a linearized model. Defining the spatial scale of a function of spatial distance as the standard deviation of the distribution obtained by scaling the function, they showed that the squared spatial scale of population density was l02 + mlg2 /γ, where l0 and lg are the scales of environmental

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noise and dispersal distance, respectively, m is the dispersal rate, and γ the

strength of local density regulation. A similar formula was given by Engen [10] for log density with larger population fluctuations but smooth migration modeled as spatial diffusion.

In addition to using linear models as approximation to more complex dy-

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namics, the result of Lande et al. [29] is based on two other simplifying as-

sumptions. First, there is no demographic stochasticity acting independently on the survival and reproduction of each individual, only spatially correlated environmental stochastisity affecting all individuals sufficiently close to one

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another in the same way. This is realistic only for large densities, or very large spatial scale of environmental effects. Secondly, the models are based on the assumption of strictly local density regulation, that is, the temporal

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changes in density at a given location is only affected by the density at that

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particular point, with no direct effect from the density at nearby locations. Here, we consider the linear model of Lande et al. [29], generalized to include

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demographic stochasticity acting locally. Furthermore, the density regulation at a given point in space is assumed to act through a given weighted average of densities in the neighborhood of the point, defining a more realistic com-

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petition for resources in space. These two additional components still leads to rather transparent analytical results and are shown to create substantial effects on the spatial autocorrelation of population density. The most common approach to studying population dynamics in space is to use so-called individual based models, simulating the dynamics based on

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the performance of each individual in relation to reproduction, survival and dispersal [46]. That method has the advantage that one can study very complex forms of spatial structure and non-linear dynamics. The drawback, on the other hand, is that one cannot obtain general analytical results but only

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investigate the performance of the system for a limited choice of parameter values compared to all realistic combinations. Another approach is to start with small cells in space and define birth- and death processes with possible

increase or decrease of a single individual at each cell during an infinitesimal time step, and allow for smooth dispersal of individuals by spatial diffu-

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sion [7, 35, 38]. Historically, this was also the approach first used to describe

stochastic dynamics of populations with no spatial structure [3], but during the 1970’ties one became aware of the practical limitations of these models due to large environmental effects affecting all individuals in the same or similar way [17, 25, 26, 34, 45, 54]. For large populations demographic noise

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which by definition is independent among individuals has practically no effect compared to environmental noise [30]. Starting with pure demographic

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noise and generalizing by including temporal environmental variation in parameters has so far not given any transparent analytical results. At the other

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extreme, the vast literature on time series analysis on log population sizes using constant variance in the noise [53] only includes the environmental component with no demographic noise and may therefor be inadequate for

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small population sizes.

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Here we follow the approach of Lande et al. [29] using the concept of population density with no reference to positioning of each individual in the population. The underlying basic idea is that individuals move around in their habitat and thereby contribute to a continuous spatial density function expressing the mean number of individuals in areas relative to these stochastic movements [13]. Demographic stochasticity is generated by within year variation among individuals in survival and reproduction [30] and thereby 6

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operates directly on the point process but indirectly also on the density function. Each death or reproduction, however, only affects the density function over a very small area where the individuals are most likely to stay. Density regulation, on the other hand, reflects the competition between individuals

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for resources spread out over larger areas where individuals search for and collect their food items. Hence, individuals with substantial difference in

mean positions may still compete with one another during foraging due to

overlapping search areas. This may in particular be the case in a seasonal

environment, where individuals from large areas may compete intensively

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during periods with limited food supplies [15, 41]. Accordingly, competition

for food creates a density regulation in such a way that the density at a given point is affected not only by the local density at that point, but also by the densities in a surrounding area where foraging is likely to occur. Here we do not model the spatial position of each individual but still take the

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independent contributions from individuals into account by adding a demographic noise component to the density, which is actually a spatial white

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noise component. This is defined in such a way that it leads to the correct magnitude of fluctuations in small areas in agreement with models without

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spatial structure [30].

Let z denote points in the two-dimensional space and write µ(z, t) for the

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population density at z at time t meaning that the mean number of individuals in a small (infinitesimal) area dz at position z at a given time t is

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µ(z, t)dz. We consider each individual to contribute to the density according to its movements in its home range so that the integral over the entire space of the density contribution from a single individual is one. The density µ(z, t) therefore is a mean value in the sense that the expected number of individuals in an area A during a discrete time step with given density field µ(z, t) is

R

A

µ(z, t)dz. Our aim is to analyze how the field µ(z, t) changes

through time by births and deaths of individuals affected by density, as well 7

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as migration. Stochastic contributions to the next generation will typically have demographic components generated by independent stochastic variation in vital rates among individuals a given year, as well as environmental components generated by a stochastic environment affecting all individuals

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at nearby locations in the same or a similar way. In simple models with

no spatial components, the relative effect of demographic and environmental

variance is determined by the population size, the demographic and environ-

mental variance components of the temporal increment in a population of size

N being proportional to N and N 2 , respectively. This variance is written as

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σd2 N + σe2 N 2 where σd2 and σe2 are the demographic and environmental vari-

ances [9]. We shall see that a similar result holds for spatial models in which the relative effects of these components in an area depend on the number of individuals expected to be in that area. One interesting question is how these variance components act together in generating the spatial autocor-

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relation function, in particular what their role is in determining the spatial scale of synchrony. Here we show that the demographic variance in relation

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to environmental variance acts in space through a spatial demographic coefficient s = σd2 /(σe2 N0 ), where N0 is a characteristic population size defined

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by the mean density and the autocovariance function for environmental noise.

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2. Deterministic model In the absence of migration and stochastic noise we assume that the density

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regulation at z acts through the weighted mean R

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µ(z − u, t)f (u)du, where

f (u) is a two-dimensional distribution obeying f (u)du = 1. The continuous model for temporal change in density then takes the form Z dµ(z, t) = rµ(z, t){1 − D[ µ(z − u, t)f (u)du]}, dt

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where D is an increasing function describing how the densities in the neighborhood of z affects the expected growth in density and r is the growth rate in the absence of density regulation. This formulation defines an overall carrying capacity K for population density defined by D(K) = 1. For a spatially

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constant population density µ(z, t) = K the growth at any point in space is accordingly zero.

Although non-linearities may play a crucial role in population dynamics [49]

a linear approximation around the carrying capacity may still be realistic for describing small fluctuations in population density [29]. The accuracy

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of this approximation depends in general on the form of density regulation

as well as the magnitude of environmental noise. Applying this small noise approximation and writing µ(z, t) = K + ε(z, t) we find to the first order in ε(z), using the fact that D(K) = 1 that

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Z dµ(z, t) = −rKD0 (K) ε(z − u, t)f (u)du, dt

or, by inserting ε(z, t) = µ(z, t) − K

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Z dµ(z, t) 2 0 0 = rK D (K) − rKD (K) µ(z − u, t)f (u)du. dt

Using for example a logistic type of density regulation, that is D(x) = x/K,

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we have D0 (K) = 1/K and

dµ(z,t) dt

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= r[K − µ(z −u, t)f (u)du]. For simplicity

of notation we write the above general linear model as

Z dµ(z, t) = α − γ µ(z − u, t)f (u)du, dt

where α = rK 2 D0 (K) and γ = rKD0 (K).

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Assuming density independent dispersal rate M the migration out of dz during dt leads to a reduction M µ(z, t)dt in density at z. We shall assume that the total dispersal M has two components, M = m + mp , where m represents local dispersal over a distance u with distribution g(u), while mp

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is panmictic long distance dispersal rate with distribution gp (u). This is a

distribution with extremely large variance that later will be considered as infinite. Migration into z then leads to an increase

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µ(z − u, t)[mg(u) +

mp gp (u)]dudt in density at location z. As a consequence, the deterministic

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model with migration is

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3. Stochastic model

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Z Z dµ(z, t) = α−γ µ(z−u, t)f (u)du−M µ(z, t)+ µ(z−u, t)[mg(u)du+mp gp (u)]du. dt (1)

For the female segment of a single closed population with no spatial structure

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the demographic variance is defined as the between-individual variation in their contribution to the next generation, that is, their number of female offspring surviving to the next census plus one if the mother survives [9].

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The environmental stochasticity is the between-year variance in the mean of these contributions, typically generated by varying environmental conditions.

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Hence, the environmental stochasticity affects by definition all individuals in the same way. With these assumptions the variance in population size N + ∆N the next year given the population size N is σd2 N + σe2 N 2 , where σd2 and σe2 are the demographic and environmental variance, respectively [9, 30]. As in Lande et al. [29] we will approximate the discrete dynamical model by a continuous one corresponding to using diffusion approximations to discrete 10

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models in the single population case. In order to illustrate the form of the environmental and demographic noise terms in a spatial model it is illustrating first to consider an area A with spatially constant density µ(z, t) = µ at time t so that the expected number of individuals is N = µA. Decom-

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posing the total noise in µ(z, t) into environmental and demographic components dµ(z, t) = dµe (z, t) + dµd (z, t) with Edµe (z, t) = Edµd (z, t) = 0,

the spatio-temporal environmental noise in the density is then given by

E[dµe (z, t)dµe (z + h, t)] = σe2 µ2 ρe (h)dt, while the demographic noise component takes the form E[µd (z, t)dµd (z + h, t)] = σd2 µδ(h)dt, where δ(h) is the ance of N + dN =

R

A

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Dirac delta function in two dimensions. With these assumptions, the variµ(z, t + dt)dz at time t + dt given that there are N

individuals in A at time t, is

Z

A

[dµd (u, t) + dµe (u, t)]du}.

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var(N + dN |N ) = var{

Applying the continuous analogy of the formula for the variance of a sum

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(see Bartlett [3] and Engen et al. [11]) we then find accordingly

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var(N + dN |N ) =

σd2 µdt

Z Z A

A

δ(u − v)dudv +

σe2 µ2 dt

Z Z A

A

ρe (u − v)dudv.

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For sufficiently small areas the environmental correlation ρe (u − v) is one for

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any two points u and v in A. Using that

R

A

δ(u − v)dv = 1 we then find that

var(N + dN |N ) = (σd2 µA + σe2 µ2 A2 )dt = (σd2 N + σe2 N 2 )dt

illustrating that the above definition of demographic and environmental noise in space is in accordance with the standard formulas in continuous time for the variance of change in population size for a single population with no 11

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spatial structure [30]. Using this formulation of the noise in the general linear model we finally obtain the stochastic spatial population model

Z

Z

µ(z − u, t)f (u)du − M µ(z, t) +

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dµ(z, t) = {α − γ

µ(z − u, t)[mg(u) + mp gp (u)]du}dt + dµe (z, t) + dµd (z, t). (2)

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Under the same assumption we can approximate µ(z, t) in the stochastic terms of equation (2) by its mean K, as done by Lande et al. [29]. For many populations the autocorrelation seems to approach a positive con-

stant for very large distances indicating a large scale common stochastic noise

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component. This noise component could be generated by fluctuations in temperature or other relevant environmental factors not varying in space. The

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environmental noise term then takes the form dµe (z, t) = dµ0 (z, t) + dµ1 (t) where the component dµ0 (z, t) has a spatial scale while dµ1 (t) is the common noise component for points in the entire space. These two components are

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assumed to be additive and independent. Writing E[dµ0 (z, t)dµ0 (z + h, t)] = µ2 σ02 ρ0 (h)dt, where ρ0 (h) is assumed to approach zero at large distances, and

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Edµ1 (t)2 = µ2 σ12 dt, the environmental correlation ρe (h) then takes the form ρe (h) =

σ02 ρ0 (h) + σ12 = ρ0 (h)(1 − ρ∞ ) + ρ∞ , σo2 + σ12

where ρ∞ = σ12 /(σ12 + σ02 ) is the correlation between noise terms at very large (infinite) distances, while the environmental variance is σe2 = σ02 + σ12 .

4. Solution for the spatio-temporal autocovariance function 12

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4.1 The Fourier transform We now assume that the parameters are chosen so that the density function is a stationary field in the entire plane with autocovariance c(h) =

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cov[µ(z, t), µ(z + h, t)]. Requiring that this covariance function is the same at time t and t + dt yields

2cov[µ(z, t), dµ(z + h, t)] + cov[dµ(z, t), dµ(z + h, t)] = 0.

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We have previously assumed small fluctuations in the derivation of the linear

form of the density regulation. Using the resulting from of equation (2) in the above equation then leads to an equation for the autocovariance function

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c(h) in the stationary model

R

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−2γ c(h − u)f (u)du − 2M c(h) + 2 c(h − u)[mg(u) + mp gp (u)]du

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+σd2 δ(h)K + σ02 ρ0 (h)K 2 + σ12 K 2 = 0.

(3)

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Writing c∗ (ω) for the Fourier transform of c(h) as defined in Appendix A and a similar notation for transforms of other functions involved, equation

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(3) takes the form

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−2γc∗ f ∗ − 2M c∗ + 2c∗ (mg ∗ + mp gp∗ ) + σd2 K + σ02 K 2 ρ∗0 + (2π)2 σ12 K 2 δ = 0,

where δ = δ(ω) is the Dirac delta function in two dimensions and ω has been omitted in all Fourier transforms to simplify the notation, giving

c∗ =

σd2 K + σ02 K 2 ρ∗0 + (2π)2 σ12 K 2 δ . 2(M + γf ∗ − mg ∗ − mp gp∗ ) 13

(4)

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In the last term of the numerator the factor δ is different from zero only for

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ω 6= 0. Further, because f , g and gp are distributions their Fourier transforms evaluated at zero are one and this term can consequently be written

separately as (2π)2 σ12 K 2 δ/(2γ). By panmictic migration we shall mean a complete spatial mixture of the individuals in the subpopulation that mi-

grates in that way. This can be achieved for example by assuming the long distance migration to have a bivariate normal distribution with zero correla-

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tion and variances approaching infinity so that the distribution approaches

a uniform distribution over the entire space. The Fourier transform gp∗ then approaches zero for ω different from zero. We shall consider models with the property that ρ∗0 , f ∗ and g ∗ all approach zero as ω1 or ω2 approaches infinity.

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By these remarks we choose to decompose equation (4) as

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c∗ = c∗0 + c∗d + c∗1 = c∗0 +

σd2 K (2π)2 σ12 K 2 δ + , 2M 2γ

(5)

where c∗d = σd2 K/(2M ) is a constant term transforming back to a function

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proportional to Dirac’s delta function δ, while c∗1 = (2π)2 σ12 K 2 δ/(2γ) is pro-

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portional to δ and transforms back to a constant. The first term

c0 ∗ =

σd2 K + σ02 K 2 ρ∗0 σd2 K − , 2(M + γf ∗ − mg ∗ ) 2M

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then approaches zero as ω1 or ω2 approaches infinity.

4.2 Stationary variance The autocovariance function c(h) can be found by the backward transforma14

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tion of c∗ given by equation (5) giving three terms c(h) = c0 (h) + cd (h) + c∞ . Here the second and third terms represent demographic and common noise, while the first term generated by the environmental and demographic noise is the term which has a spatial scale and approaches zero at large distances.

v = v0 + vd + v1 .

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Plugging in h = 0 then yields correspondingly three variance components

The backward transformation of the constant c∗d = σd2 K/(2M ) is δ(z)σd2 K/(2M )

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showing that this is a white noise term with no spatial autocorrelation but infinite variance vd = ∞. Consider therefore instead the mean density over a small area A, µ ¯ = A−1

R

A

µ(z)dz. This demographic variance component is

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then proportional to 1/A,

σd2 K . 2M A

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vard (¯ µ) =

The last variance term representing the common noise yields a covariance c1 (h) = v1 = σ12 K 2 /(2γ). The corresponding variance component for the

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mean density over a small area A is v1 , independent of A.

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The first variance components can only be found by performing the backwards transformation numerically as explained in Appendix A and illustrated

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in Figs. 1-3., which is a simple one-dimensional integration in the isotropic case. This is the only term with a spatial scale which we analyze below.

4.3 Spatial scale of population synchrony Now consider the spatial scale of the term c0 generated by environmental and demographic noise, as well as migration and density regulation. We will 15

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show that the effect of the demographic variance on this scale is determined by the function σd2 σe2 N0

where N0 = K

R

R2

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s=

ρ0 (z)dz = KA0 and R2 denotes the entire two-dimensional

space. A function which is one in some area A0 , that we call the character-

istic area, and otherwise zero will have the same integral (volume) over R2 .

The quantity N0 = KA0 is the expected number of individuals in A0 and s is

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the ratio between the demographic term σd2 N0 and the environmental term σe2 N02 of the noise affecting the mean number of individuals in this area. Ac-

cordingly we call s the spatial demographic coefficient as it actually expresses the effect of the demographic variance relative to the environmental variance in spatial population dynamics. Notice that this parameter is independent

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of the scale of measurement used for distance and area (for example m2 or km2 ) so it is a unique measurement of the relative effect of demographic and

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environmental stochasticity in space.

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Following Lande et al. [29] we define the spatial scale l of the autocovariance function c(h) or autocorrelation ρ(h) in a given direction as the standard deviation of the marginal distribution along that direction defined by scaling

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c(h) or ρ(h) to integrate to one. Defining correspondingly l0 , lf and lg derived from ρ0 , f and g and applying the general scaling formula given in Appendix

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B to c∗ (ω) then gives the squared spatial scale of c0 as "

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(1 + s)(mlg2 − γlf2 ) M 2 l = l + . M + s(m − γ) 0 M +γ−m 2

(7)

In this model the function c0 (h) may actually take negative values even if ρ0 (h) is non-negative due to the effect of spatial density regulation. In that 16

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case l2 can no longer strictly be interpreted as a variance. However, the above relation is still very informative in expressing how the scale l changes by minor deviations from the model considered by Lande et al. [29] due to demographic noise and density regulation acting in space as expressed by the

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function f . For a more complete analysis one will have to perform the inverse transformation of c∗0 numerically (Figs 1-3). In the isotropic case this can be done by a single one-dimensional integration as explained in Appendix A.

We arrive at the simple formula l2 = l02 + lg2 m/γ derived by Lande et al. [29] under no panmictic migration, by plugging in s = 0 for no demographic

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stochasticity, M = m, and lf = 0 for no spatial scale of density regulation.

Actually, adding only panmictic migration to the model of Lande et al. [29] will have the same effect as increasing the strength of density regulation from γ to M + γ − m = γ + mp .

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Adding only demographic stochasticity to this simplest form of the model

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with no panmictic migration yields

1 [l02 + lg2 (1 + s)M/γ] 1 + s(1 − γ/M )

(8)

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l2 =

showing that the effect of migration relative to that of environmental noise

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increases with demographic stochasticity by the factor (1 + s). If there is no panmictic migration and no demographic variance the formula

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takes the form

l2 = l02 + lg2 M/γ − lf2

(9)

showing that increasing spatial scale of density regulation, for example by increasing foraging areas, tends to decrease the spatial scale of population 17

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synchrony.

4.4 Spatial autocorrelation in counts

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As mentioned earlier, the expected number of individuals in an area at a given time is the mean density over that area, while the actual number of individuals is stochastic depending on the particular movement and position-

ing of each individual. When sampling an area A a variance component in addition to the temporal variance of density must therefore be added. This

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also applies to traps sampling an extremely small area but operating through some time interval during which moving individuals are caught. The simplest point process describing positioning of individuals is the inhomogeneous Poisson process giving a so-called Cox process [8] for the total

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counts when the stochasticity in µ(z, t) is also taken into account. For this model the counts conditioned on density are Poisson distributed with mean

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value equal to the mean density and variance equal to the mean. For small fluctuations in density this yields an additional variance KA for counts in an area A small enough for the density to be approximately constant over the

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area A. For more complex point patterns with overdispersion due to clumping or underdispersion due do competition, the variance will be νKA where

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ν is a measurement of overdispersion which is 1 for the Poisson process [13]. The additional term due to sampling, however, is only relevant in relation

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to sampling and does not affect the dynamics of the field µ(z, t). Since we also have a demographic variance term in µ(z, t) due to independent birth

and deaths of individuals, the total demographic variance term in the counts σ2

will be ( 2Md + ν)KA, which means that the realized demographic variance for counts in small areas with approximately constant density must be

σd2 2M

+ ν.

The variance in the count N in an area A, small enough for µ(z, t) to be

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approximately constant over the area, is V (A) = var(µA) + νKA, which in terms of the covariance function is

σd2 + ν)KA + (c0 (0) + c∞ )A2 . 2M

(10)

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V (A) = (

The covariance between independent counts in two areas A1 and A2 at distance h measured from their midpoints is then approximately

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C(h; A1 , A2 ) = (c0 (h) + c∞ )A1 A2

(11)

and the corresponding correlation is

(12)

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C(h; A1 , A2 ) . R(h; A1 , A2 ) = q V (A1 )V (A2 )

The effect of an increase in the spatial scale of density regulation leading to

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decrease in spatial synchrony and even to a negative autocorrelation in the simplest case of no demographic variance is illustrated in Fig. 1. This can be

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understood intuitively because increased spatial scale of density regulation may correspond to individuals having larger home ranges so that there is

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unlikely that two individuals are positioned very close to one another with the consequence that densities at close distances (within home ranges) may

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be negative. The effect of varying the demographic variance when the spatial scale of density regulation is zero is exemplified in Fig. 2. In equation (11) the areas may be replaced by the EN/K. Then, because c0 (0) + c∞ appearing in C(h; A1 , A2 ) is proportional to K 2 , we see that the correlation is independent of the scale of measurement, only depending on the scale through the expectations EN . Choosing a small h so that c0 (h) ≈ c0 (0) we see that R(h, A, A) actually approaches zero as A tends to zero. This 19

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means that the correlation function R(h, A, A) near h = 0 depends strongly on A. Estimating the autocorrelation from data one usually finds a sudden decrease from R(0, A, A) = 1 to R(h, A, A) smaller than one for a small h. We see that this decrease depends strongly on the sampling area (or by

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analogy the time a trap is operating) and will therefore in practice be quite difficult to interpret because it is a joint effect of sampling and demographic stochasticity. A sudden decrease to rather small correlations, for example,

may not imply that the spatial autocorrelations at larger distances or between large areas are of little importance. It may just result from choosing

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small sampling areas so that the demographic component generated by the individual point process and demographic stochasticity gets large compared to the contributions from a fluctuating environment. Theoretically, if the sampling area approaches zero the correlation drops immediately from one to zero. The effect of the sampling area on the correlation R(0+ ; A, A) which

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is the limit of R(h, A, A) as h approaches zero, is illustrated in Fig. 3 for different standard deviations lg of migration distance in an example with a

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positive spatial scale of density regulation as well as a positive demographic

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variance.

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5. Discussion

Intuitively, one important effect of random dispersal is to smooth the spatial fluctuations in population density and hence reduce the spatial variance.

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Another effect, which is not so easily understood is the effect of dispersal on the synchrony of population fluctuation. How, and at which spatial scale does dispersal affect population synchrony? Lande et al. [29] derived a remarkably simple formula that gives important general quantitative insight into this problem, although their model was based on several simplifying assumptions. Adopting the standard deviation l of the scaled autocorrelation 20

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function in a given direction as measurement of spatial scale, they showed that l2 = le2 + lg2 m/γ, where the first term represents the Moran effect of spatial correlation in the noise while the last term is the effect of a dispersal rate m and standard deviation lg of dispersal distance when the local strength

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of density regulation is γ. The most striking observation from this formula is that even small migration rates over small distances may have large effects on the scale of population synchrony if the local density regulation is

weak. Although this result is based on linearization of the dynamics under small fluctuations, Engen [10] showed that the same formula was valid for

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log population sizes under large fluctuation when dispersal was modeled as diffusions in continuous time. This conclusion agrees well with the large dif-

ferences found between the scale of oceanic fish species with extremely large scale of more than 500 kilometers [39] and butterflies [21, 48] with a scale at the order a few kilometers [22, 32]. Birds [14, 49, 50] and mammals [19, 20]

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take an intermediate position with values from 30 to 200 kilometers [15, 33]. Empirical studies usually find that the spatial autocorrelations approaches

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some positive constant at very large distances [14, 19, 50]. This is an effect of a stochastic environmental component which is the same over the entire

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field, here modeled by a positive component ρ∞ . This component generates a common component of the variance in density proportional to σe2 ρ∞ and

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decreases by increasing density regulation and panmictic migration. Here we use the same linear model as in Lande et al. [29] but have in addi-

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tion analyzed the effects of two biological factors that commonly represent important components of the dynamics. First, we have added demographic noise to the local process. It is important to make a clear distinction between demographic noise in the dynamics of density and the demographic effect of individuals considered as points in space that may or may not appear in samples. The distinction is easiest seen by considering an area A 21

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with approximately constant density µ and expected individual number µA at a given time. Then the environmental and demographic stochasticity refer to the temporal change in µ generated by vital rates of individuals. If N , the number of individuals in A, during time interval ∆t = 1 changes

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to λN , then environmental and demographic noise refers to the variance var(λ) = σe2 +σd2 /N , or var(∆N ) = σe2 N 2 +σd2 N . The density correspondingly

changes by the same stochastic factor λ during the same time interval. In a spatial model the environmental component σe2 will have a spatial structure with spatial scale le , while the demographic noise referring to independent

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variation in birth and deaths among individuals is practically independent even for very small distances. We have argued that this, under small fluctuations in density, can be modeled by the spatial noise term dµd (z, t) with the property E[dµd (z, t)dµd (z + h, t)] = σd2 Kδ(h)dt, while the environmental

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noise takes the form E[dµe (z, t)dµe (z + h, t)] = σe2 K 2 ρe (h)dt. In a population with no spatial effects the demographic and environmental variance terms are equal in size if N = N1 = σd2 /σe2 , a ratio that often is at the

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order of say 50 to 200. In our spatial model there are no restrictions on the size of the total area which is actually considered to be infinite. So, the de-

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mographic variance is of no interest with respect to the total population size, but it is still of interest when analyzing how environmental and demographic

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noise act together in specified areas or at given distances. We have found that the individual number N0 = K

R

R2

ρ0 (z)dz = KA0 plays an important

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role in quantifying how the ratio between environmental and demographic variance affects the synchrony of population fluctuations through the spatial demographic coefficient s = σd2 /(N0 σe2 ) = N1 /N0 . Hence, s is large if N0 is small compared to N1 which occurs if the non-constant component of the environmental autocorrelation has small scale so that the environmental correlation component ρ0 (h) is close to zero at small distances. If this autocorrelation decreases very slowly with distance, then the demographic noise 22

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has little effect on the spatial scale of the density. The characteristic area A0 , determined by the environmental autocorrelation, may be viewed as the typical area in which there is an interaction between the demographic and environmental stochasticity which partly determines spatial scaling. Adding

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only demographic noise to the model of Lande et al. [29] yields the scaling

result given by equation (8) under no panmictic migration, indicating that

the effect of dispersal relative to the Moran effect is proportional to s + 1. However, the first factor in this expression indicates that demographic noise affects the total scaling in a more complex way.

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Equation (5) shows that demographic noise adds a white noise term δ(z)σd2 K/(2M ) to the density field corresponding approximately to a variance σd2 N/(2M ) in an area with N individuals. It appears that this effect on the variance is large for small migration rates and vanish for very large values of M . If there

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is little dispersal or distances moved are very short, then the large effect of demographic noise in small areas adds trough time as new generations tend to stay close to where each individual was born which in turn adds new lo-

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cal stochasticity by the demographic noise of their vital rates. Dispersal, on the other hand, smooths this out in space and reduces the spatial effects of

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correlated noise and weak density regulation.

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Our second generalization yields the way density regulation is modeled. In the model of Lande et al. [29] density regulation acts at each point with no influence from density at locations in the neighborhood. However, even species

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that are not territorial usually compete for resources in some restricted home range with size varying a lot among species. Therefore, individuals are not only affected by the density at the center of their home range but in all of it, probably with weaker dependence at larger distance from the center. The territory of an individual is the space it defends against intruders of the same species and is smaller than the home range. For distances within the 23

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territory the density regulation is very strong. Accordingly we have modeled the density regulation at location z by a function D of a weighted mean R

µ(z − u, t)f (u)du of densities, where f is a two dimensional distribution

with standard deviation lf in some specified direction of interest. Thus our

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approach is formulated through assumptions on how the field of densities

changes through time by averaging over movements of individuals in their

home range during some discrete time interval (such as one year) without trying to incorporate the actual positioning of individuals. This is a simplification that leads to rather transparent results relative to approaches based

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on modeling positioning and interactions between individual (see Ovaskainen

and Cornell [38] with references for individual based models with spatial interactions). However, our model is still based on linearization of the dynamics around the equilibrium value, which requires that fluctuations in density

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around the equilibrium are moderate or small [30].

The effect of lf ignoring the effect of demographic stochasticity is simply the term lf2 subtracted from l2 in equation (9). In Fig. 3 this effect is

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demonstrated graphically also showing its effect on the whole autocorrelation function. It appears that the spatial scale of density regulation may have

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considerable effect on the autocorrelation and can even make the correlation negative even if the environmental noise only has positive correlations (Fig.

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3).

With modern GPS based telemetry used to track the movements of individ-

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uals combined with new statistical techniques for analyzing such data [37] knowledge of home range and territory sizes is accumulating rapidly [15, 41] . The present theory may prove useful in utilizing this knowledge to analyze in some detail for different types of species how density regulation acts in space and affects the synchrony of population fluctuations.

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Acknowledgements This study was financed by the European Research Council (ERC-2010-AdG

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268562) and the Research Council of Norway (SFF-III 223257/F50).

Appendix A Fourier transforms

defined as c∗ (ω) = c∗ (ω1 , ω2 ) =

by

Z Z

ei(ω1 h1 +ω2 h2 ) c(h1 , h2 )dh1 dh2

√ −1 is the imaginary unit. The inverse transformation is given

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where i =

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The Fourier transform of the autocorrelation function c(h) = c(h1 , h2 ) is

1 Z Z −i(ω1 h1 +ω2 h2 ) ∗ e c (ω1 , ω2 )dω1 dω2 . (2π)2

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c(h) =

q

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If the function c(h1 , h2 ) is isotropic, that is, it depends on h only through r = h21 + h22 , then c∗ is also isotropic depending on ω through u =

q

ω12 + ω22 . In

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this case, writing the functions as c(r) and c∗ (u), the inverse transformation

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can be written as a one-dimensional integral

c(r) =

1 Z∞ ∗ c (u)J0 (ru)udu 2π 0

where J0 is the Bessel function of the first kind of order zero [1]. For an 2

2

2

isotropic function of the Gaussian form c(h) = σ 2 e−(h1 +h2 )/(2l ) , the Fourier 2 l2 /2

transform is c∗ (u) = 2πl2 σ 2 e−u

.

25

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Appendix B Spatial scale of synchrony If the covariance function is scaled so that it integrates to one, the corre-

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sponding Fourier transform is the characteristic function of that distribution multiplied by a constant factor. Hence, the log of this c∗ is, apart from a constant added term, the corresponding cumulant generating function. Therefore, the squared spatial scale in the direction of the first axis, defined as the

l2 = −

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second cumulant (the variance), is simply

∂2 ln c∗ (ω1 , ω2 )|ω1 =ω2 =0 . ∂ω12

Applying this to the Fourier transform given by equation (6) then yields the

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general scaling result given by equation (7).

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Legends Fig. 1. The effect of spatial scale of density regulation The upper panel shows the spatial correlation between independent counts in

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two areas of size A, R(h; A, A), as function of distance h for different values

of the scale of density regulation lf when the sampling areas are A = 0.3

and there is no demographic variance (σd2 = 0) and no panmictic migration (mp = 0). The other parameters are le2 = 100, lg2 = 2, γ = 0.1, σe2 = 0.01, mp = 0, M = 0.2, ρ∞ = 0.2, ν = 1 and K = 1. The lower panel shows the

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spatial scale l as function of lf given by equation (9) for the same values of the other parameters.

Fig.2. The effect of the spatial demographic coefficient The upper panel shows the spatial correlation R(h; A, A) as function of dis-

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tance z for different values of the spatial demographic coefficient s when sampling areas are A = 0.3, there is only local density regulation (lf = 0)

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and no panmictic migration (mp = 0). The other parameters are le2 = 100, lg2 = 2, γ = 0.1, σe2 = 0.01, mp = 0, M = 0.2, ρ∞ = 0.2, ν = 1 and

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K = 1. The lower panel shows the corresponding spatial scale l as function of s given by equation (8) for the same values of the other parameters. With

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these parameter values σd2 ≈ 5s. Fig. 3. The effect of sampling area

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The correlation R(0+ ; A, A) as function of A for different values of the standard deviation lg of the migration distance. The other parameters are are

mp = 0, le2 = 100, σe2 = 0.01, σd2 = 0.8, γ = 0.1, lf = 5, M = 0.2, ρ∞ = 0.2, ν = 1 and K = 1.

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0.16

lf = 10

0.10

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0.12

5

0.08

0

0.06 0.04 0.02

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Autocorrelation

0.14

0.00 -0.02 0

10

20

30

40

M

Distance

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10

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8 6 4

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Spatial scale l

12

2

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0

0

1

2

3

4

5

6

7

8

Scale of density regulation lf

fig.1 34

9

10

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0.07

0

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s=

0.05

04

0.

0.04

08

0.

0.03 0.02

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16

0.

Autocorrelation

0.06

0.01 0.00 0

10

20

30

40

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Distance

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10.0

9.6

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9.4

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9.8

9.2

9.0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20

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Spatial scale l

10.2

Spatial demographic coefficient s

fig.2 35

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1.0 0.9

4

0.7

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0.6 0.5 0.4 0.3

0.1

0.4

AC

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0.2

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0.2

0.0 0.0

3

2

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Correlation R(0+; A,A)

0.8

0.6

0.8

lg = 1

1.0

1.2

Sampling area A

fig.3

36

1.4

1.6

1.8

2.0